The Chain Rule

As a motivation for the chain rule, consider the function

f(x) = (1+x2)10.

Since f(x) is a polynomial function, we know from previous
pages that f'(x) exists. Naturally one may ask for an explicit
formula for it. One tedious way to do this is to develop
(1+x2)10 using the Binomial Formula and then take the
derivative. Of course, it is possible to do this, but it won't
be much fun. But what if we have to deal with
(1+x2)100!
Then I hope you agree that the Binomial Formula is not the way to
go anymore.

So what do we do? The answer is given by the Chain Rule.
Before we discuss the Chain Rule formula, let us give another
example.

Example. Let us find the derivative of
.
One way to do that is through some trigonometric identities.
Indeed, we have

So we will use the product formula to get

which implies

Using the trigonometric formula
,
we get

Once this is done, you may ask about the derivative of
? The answer can be found using similar trigonometric
identities, but the calculations are not as easy as before. Again
we will see how the Chain Rule formula will answer this question
in an elegant way.

In both examples, the function f(x) may be viewed as:

where
g(x) = 1+x2 and
h(x) = x10 in the first example, and
and g(x) = 2x in the second. We say that
f(x) is the composition of the functions g(x) and
h(x) and write

The derivative of the composition is given by the formula

Another way to write this formula is

where
and u = g(x). This second
formulation (due to Leibniz) is easier to remember and is the
formulation used almost exclusively by physicists.

Example. Let us find the derivative of

We have
,
where
g(x) = 1+x2 and
h(x) =
x100. Then the Chain rule implies that f'(x) exists, which
we knew since it is a polynomial function, and

Example. Let us find the derivative of

We have
,
where g(x) = 5x and
.
Then the Chain rule implies that f'(x) exists and

In fact, this is a particular case of the following formula

The following formulas come in handy in many areas of techniques
of integration.